Efficient and accurate numerical quadrature for immersed boundary methods

Kudela, László; Zander, Nils; Bog, Tino; Kollmannsberger, Stefan; Rank, Ernst

doi:10.1186/s40323-015-0031-y

Cited by 80 publications

(58 citation statements)

References 34 publications

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“…For the case of a void local model, we proposed the simple construction of an exact NURBS domain to fill in the geometric detail by adding multiple interpolatory control points at the centre. The procedure applies directly to all types of star domains and may require a few additional improvements (such as based on very recent works [44,45]) in the general case. For the situation of covered local models, the quadrature rule coming from the local problem was transposed within the global NURBS patch.…”

Section: Resultsmentioning

confidence: 99%

“…Corresponding investigations are in progress to generalize the procedure. We also note that new strategies producing conforming quadrature rules for trimmed surfaces have appeared very recently (see Nagy et al [44] and Kudela et al [45]) and may constitute promising options to be considered in our future works.…”

Section: Implementation: Computation Of the Interface Reaction Forcesmentioning

confidence: 99%

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Local enrichment of NURBS patches using a non-intrusive coupling strategy: Geometric details, local refinement, inclusion, fracture

Bouclier

Passieux

Salaün

2016

Computer Methods in Applied Mechanics and Engineering

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OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. This is an author-deposited version published in: http://oatao.univ-toulouse.fr/ Eprints ID : 14633 Highlights• We address the issue of modelling local phenomena in a NURBS patch with a global/local non-intrusive algorithm.• The approach provides simplicity and flexibility to treat any case of local enrichment in a NURBS patch.• We propose a strategy to handle non-conforming geometries.• We demonstrate the performance of the method on a range of two-dimensional linear elastic numerical examples. AbstractIn this work, we apply a non-intrusive global/local coupling strategy for the modelling of local phenomena in a NURBS patch. The idea is to consider the NURBS patch to be enriched as the global model. This results in a simple, flexible strategy: first, the global NURBS patch remains unchanged, which completely eliminates the need for costly re-parametrization procedures (even if the local domain is expected to evolve); then, easy merging of a linear NURBS code with any other existing robust codes suitable for the modelling of complex local behaviour is possible. The price to pay is the number of iterations of the non-intrusive solver but we show that this can be strongly reduced by means of acceleration techniques. The main development for NURBS is to be able to handle non-conforming geometries. Only slight changes in the implementation process, including the setting up of suitable quadrature rules for the evaluation of the interface reaction forces, are made in response to this issue. A range of numerical examples in two-dimensional linear elasticity are given to demonstrate the performance of the proposed methodology and its significant potential to treat any case of local enrichment in a NURBS patch simply.

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Section: Resultsmentioning

confidence: 99%

Section: Implementation: Computation Of the Interface Reaction Forcesmentioning

confidence: 99%

Local enrichment of NURBS patches using a non-intrusive coupling strategy: Geometric details, local refinement, inclusion, fracture

Bouclier

Passieux

Salaün

2016

Computer Methods in Applied Mechanics and Engineering

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“…A second idea could be to use the divergence theorem to transform the domain integral to a boundary integral [112][113][114]. The third option is proposed by Kudela et al [115,116] for two-dimensional structures and has been extended to three-dimensional problems by Dumonet [117]. They propose that the cells that are intersected by the physical boundary be divided into a set of curved triangles and quadrilaterals.…”

Section: A Second Mapping From the Canonical Polygonal Elementmentioning

confidence: 98%

The finite cell method for polygonal meshes: poly-FCM

Duczek

Gabbert

2016

Comput Mech

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In the current article, we extend the two-dimensional version of the finite cell method (FCM), which has so far only been used for structured quadrilateral meshes, to unstructured polygonal discretizations. Therefore, the adaptive quadtree-based numerical integration technique is reformulated and the notion of generalized barycentric coordinates is introduced. We show that the resulting polygonal (poly-)FCM approach retains the optimal rates of convergence if and only if the geometry of the structure is adequately resolved. The main advantage of the proposed method is that it inherits the ability of polygonal finite elements for local mesh refinement and for the construction of transition elements (e.g. conforming quadtree meshes without hanging nodes). These properties along with the performance of the poly-FCM are illustrated by means of several benchmark problems for both static and dynamic cases.

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“…More sophisticated integration schemes have been developed to efficiently evaluate the discontinuous domain integrals for the FCM. These schemes include the blended partitioning using the smart octree, moment fitting, adaptively weighted quadratures, and quadratic reparametrization for the tetrahedral FCM …”

Section: The Fcm For Embedded Interface Problemsmentioning

confidence: 99%

Multi‐level hp‐finite cell method for embedded interface problems with application in biomechanics

Elhaddad

Zander

Bog

et al. 2018

Numer Methods Biomed Eng

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This work presents a numerical discretization technique for solving 3-dimensional material interface problems involving complex geometry without conforming mesh generation. The finite cell method (FCM), which is a high-order fictitious domain approach, is used for the numerical approximation of the solution without a boundary-conforming mesh. Weak discontinuities at material interfaces are resolved by using separate FCM meshes for each material sub-domain and weakly enforcing the interface conditions between the different meshes. Additionally, a recently developed hierarchical hp-refinement scheme is used to locally refine the FCM meshes to resolve singularities and local solution features at the interfaces. Thereby, higher convergence rates are achievable for nonsmooth problems. A series of numerical experiments with 2- and 3-dimensional benchmark problems is presented, showing that the proposed hp-refinement scheme in conjunction with the weak enforcement of the interface conditions leads to a significant improvement of the convergence rates, even in the presence of singularities. Finally, the proposed technique is applied to simulate a vertebra-implant model. The application showcases the method's potential as an accurate simulation tool for biomechanical problems involving complex geometry, and it demonstrates its flexibility in dealing with different types of geometric description.

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Efficient and accurate numerical quadrature for immersed boundary methods

Cited by 80 publications

References 34 publications

Local enrichment of NURBS patches using a non-intrusive coupling strategy: Geometric details, local refinement, inclusion, fracture

Local enrichment of NURBS patches using a non-intrusive coupling strategy: Geometric details, local refinement, inclusion, fracture

The finite cell method for polygonal meshes: poly-FCM

Multi‐level hp‐finite cell method for embedded interface problems with application in biomechanics

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